Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 104

tive assessment could be made as to which metabolites do and which do not
have normal concentrations. This together with the presumed availability of
metabolite concentrations for the corresponding healthy tissues represent the
critical step towards decision making about disease assignment. In aiding
these clinical decisions, the mentioned advanced mathematical methods must
unambiguously separate the physical from nonphysical (noise, noiselike) con
tents in the input timedomain data, to reconstruct exactly the true number
of individual resonances and, finally, to deduce the concentrations of every
genuine metabolite.
The signal processor capable of fulfilling all these most stringent physical
as well as clinical criteria is the fast Pade transform. This method yields the
unique component shape spectra as on panel (iii) i n Fig. 3.2 that are generated
from the exact envelope. This will be thoroughly documented in the present
chapter through maximally accurate reconstructions of all the physical spec
tral parameters, including their number K. Further, we shall implement exact
signalnoise separation by reliance upon the concept of Froissart doublets [44]
or polezero cancellations [45].

3.2 Absorption total shape spectra
3.2.1 Absorption total shape spectra or envelopes
Figure 3.3 shows, on panels (i) and (ii), the real Re(c n ) and imaginary Im(c n )
part, respectively, of the complexvalued synthesized time signal{c n }(0≤n≤
N−1) computed from (3.1) by using the spectral parameters given in Table
3.1 . The full signal length of this MR time signal is N = 1024, the selected
bandwidth is 1000 Hz, such that the sampling rate is τ = 1 ms, and the total
duration time is T = Nτ = 1.024 s. Panel (iii) of Fig. 3.3 displays the initial
convergence regions of the FPT (+) and FPT (−) located inside and outside the
unit circle|z|< 1 and|z|> 1 in the complex planes of the harmonic variables
z and z −1 , respectively.
Since the Pade spectra are rational functions given by the quotients of two
polynomials, the Cauchy analytical continuation principle lifts the restrictions
of the initial convergence regions. Namely, the Cauchy principle extends the
initial convergence region from|z|< 1 to|z|> 1 for the FPT (+) and similarly
from |z|> 1 to |z| < 1 for the FPT (−) . Thus, both FPT (+) and FPT (−)
continue to be computable throughout the complex frequency plane without
encountering any divergent regions. An exception is the set of the fundamental
frequencies of the examined FID that are simultaneously the singular points
(poles) of the system's response function.
The small dots seen on panel (iii) depict both the exact input harmonic
variables z ±1
k

= exp (±iω k τ) and the corresponding Pade counterparts z ± k

=